A185221 E.g.f. is solution to y = 1 + log(1 + x*y) in powers of x.
1, 1, 1, -1, -10, -6, 294, 1350, -14624, -197568, 703800, 34790040, 100585968, -7259053296, -85604489712, 1588693382640, 46549054391040, -216669088277760, -24865626969568512, -159153249738896640, 13379663931502199040
Offset: 0
Keywords
Examples
y = 1 + x + 1/2*x^2 - 1/6*x^3 - 5/12*x^4 - 1/20*x^5 + 49/120*x^6 + 15/56*x^7 + ...
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..200
Programs
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Maxima
a(n):=if n=0 then 1 else sum(binomial(n+k+1,n) * sum((-1)^(j) * binomial(k+1,j) * sum((-1)^i * i! * binomial(j+i-1,j-1) * stirling1(n,i), i,1,n), j,1,k+1), k,0,n) / (n+1); /* Vladimir Kruchinin, Mar 29 2013 */
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PARI
{a(n) = local(A); if( n<0, 0, A = 1 + O(x); for( k=1, n, A = 1 + log(1 + x * A)); n! * polcoeff( A, n))}
Formula
E.g.f. is solution to y = y' * (1 - x + x*y).
a(n) = sum(k=0..n, binomial(n+k+1,n) * sum(j=1..k+1, (-1)^(j) * binomial(k+1,j) * sum(i=1..n, (-1)^i * i! * binomial(j+i-1,j-1) * stirling1(n,i)))) / (n+1), n>0, a(0)=1. [Vladimir Kruchinin, Mar 29 2013]
Lim sup n->infinity (|a(n)|/n!)^(1/n) = abs(LambertW(-1)) = 1.37455701074370748653... (see A238274). - Vaclav Kotesovec, Feb 24 2014
a(n) = n! * Sum_{k=0..n} Stirling1(n,k)/(n-k+1)!. - Seiichi Manyama, Nov 07 2023